Coding up a simple geometric algebra in MATLAB

[hunt_mat]

Hi,

I have been wanting to do this for a while but not too sure how to go about it. I have the following geometric algebra
$\lbrace\mathbf{e}_{i}\rbrace_{i=0}^{3}$ which satisfy the following relations: $$\mathbf{e}_{i}\mathbf{e}_{j}=-\mathbf{e}_{j}\mathbf{e}_{i}$$ and $$\mathbf{e}_{1}^{2}=\mathbf{e}_{2}^{2}=\mathbf{e}_{3}^{3}=1\quad \mathbf{e}_{0}^{2}=\frac{1}{\varepsilon}$$

There are 16 elements in this geometric algebra. I thought about doing it as one long vector but didn't know if there was a better way of doing it. I also am not quite sure about dealing with the $\varepsilon$, any suggestions?

Mat

 

[kreil]

You could take an object-oriented approach, like this guy did:

http://www.cgl.uwaterloo.ca/~smann/GABLE/tutorial.pdf

Of course this requires overloading any MATLAB functions that you would want to use with the objects (+, -, *, <, >, norm, exp, etc...)

 

[hunt_mat]

I am aware of GABLE but it's not the geometric algebra which I am interested in.

 

[kreil]

I am aware of GABLE but it's not the geometric algebra which I am interested in.

I wasn't really suggesting you use GABLE, but that you use the same type of object oriented approach that was used to create GABLE.

You can write a class that creates objects of the geometric algebra with all of the properties you listed.

Here is another example that implements a Clifford Alebra using an object oriented approach:

http://www.mathworks.com/matlabcentral/fileexchange/34286-clifford-algebra

 

[hunt_mat]

Anyone else care to comment?

 

[Number Nine]

I think Kriel's approach is the right one. You could represent multivectors as ordinary 1-D arrays, and then write functions to work with them, but an object oriented approach seems like the most user-friendly way to go about it.

 

[hunt_mat]

The trick comes in with how to represent epsilon in te code which I have no idea what to do with it.

I've not done much OO, and NONE with matlab.